ON THE TOPOLOGICAL DIMENSION OF THE SOLUTIONS SETS FOR SOME CLASSES
OF OPERATOR AND DIFFERENTIAL INCLUSIONS Ralf Bader
Technical University Muenchen D-80290 Muenchen, Germany e-mail: bader@appl-math.tu-muenchen.de Boris D. Gel’man1 Mikhail Kamenskii1 Valeri Obukhovskii1 Department of Mathematics Voronezh University (394693) Voronezh, Russia e-mail: gelman@func.vsu.ru e-mail: Mikhail@kam.vsu.ru e-mail: valeri@ob.vsu.ru Abstract
In the present paper, we give the lower estimation for the topo-logical dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form F = SPF where PF is
the superposition multioperator generated by the Carath´eodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topo-logical dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.
1The work of B.Gelman, M.Kamenskii and V.Obukhovskii is partially supported by the
Keywords and phrases: solutions set, fixed points set, topologi-cal dimension, multivalued map, condensing map, topologitopologi-cal degree, differential inclusion, periodic problem.
2000 Mathematics Subject Classification: 34A60, 34C25, 34C99, 47H04, 47H09, 47H11, 54F45.
1. Introduction
The investigation of topological properties of solutions sets of operator and differential inclusions in abstract spaces attracts the attention of many re-searchers (see, e.g. the recent monograph [8]). The topological dimension of solutions sets was studied in the papers [19, 9, 18, 11, 12] and others.
In the present paper, we develop the results of [11, 12] and give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space (Theorem 2.3). The abstract result is applied to the fixed points set of the multioperator of the form F = SPF where PF is the superposition multioperator generated by the Carath´eodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension (Theorem 3.4). In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space (Theorems 4.3 and 4.7).
2. Topological dimension of the fixed points set for a multimap
Let E be a Banach space; by the symbol 2E we denote the collection of all
subsets of E;
P (E) = {Ω ∈ 2E : Ω is nonempty},
Kv(E) = {Ω ∈ P (E) : Ω is convex compact}. Let us recall some notions (see, e.g. [3, 4, 13, 15, 14]). Let (A, ≥) be a partially ordered set. A map
is called the measure of noncompactness (MNC) in E if β(coΩ) = β(Ω)
for every Ω ∈ 2E.
A MNC β is called:
(i) monotone if Ω0, Ω1∈ 2E, Ω0 ⊂ Ω1 implies β(Ω0) ≤ β(Ω1);
(ii) nonsingular if β({a} ∪ Ω) = β(Ω) for every a ∈ E, Ω ∈ 2E.
As an example of MNC possessing all these properties we may consider the Hausdorff MNC,
χ(Ω) = inf{ε > 0 : Ω has a finite ε − net}.
In the sequel, we will consider only monotone nonsingular measures of non-compactness.
Definition 2.1. Let X be a closed subset of a Banach space E, L a compact topological space. A multimap F : X → Kv(E) or a family of multimaps G : X × L → Kv(E) is said to be condensing with respect to an MNC β (or simply β-condensing) if
β(F(Ω)) ≥ β(Ω) or, respectively,
β(G(Ω × L)) ≥ β(Ω) imply the relative compactness of Ω for every Ω ⊂ X.
Below we deal with the classes of upper semicontinuous (u.s.c.), lower semicontinuous (l.s.c.), and continuous multimaps (necessary definitions and detailes may be found in the same sources).
Now, let U ⊂ E be a bounded open set and F : U → Kv(E) be an u.s.c. β-condensing multimap. A multimap Φ = i−F : U → Kv(E) defined by the formula Φ(x) = x − F(x) will be called a β-condensing multifield generated by F.
It is known (see, e.g. [3, 4, 14]) that for every multifield Φ = i − F ∈ D(U , ∂U ) the topological degree deg(Φ, U ) is defined. This integer charac-teristic possesses properties which are standard for the topological degree theory.
Let us denote by N (Φ, U ) the set of all nonsingular points of Φ which coincides with the fixed points set of F, i.e.
N (Φ, U ) = { x ∈ U : x ∈ F(x)} = { x ∈ U : 0 ∈ Φ(x)}.
It easy to prove that the set N (Φ, U ) is compact. We will study the topo-logical dimension dim (see, e.g., [1, 10]) of this set.
The following statement will play an important role in our constructions. Lemma 2.2 ([19]). Let X be a paracompact topological space, dim(X) ≤ n − 1 where n ≥ 1. Let E be a Banach space and T : X → Kv(E) a l.s.c. multimap satisfying the following conditions:
(i) 0 ∈ T (x) for all x ∈ X; (ii) dim(T (x)) ≥ n for all x ∈ X.
Then there exists a continuous selection f : X → E of T such that f (x) 6= 0 for all x ∈ X.
Using this result we may give the following ”condensing” version of Theorem 2.2 of [12] giving the lower estimate for the dimension of the set N (Φ, U ). Theorem 2.3. Let Φ = i − F ∈ DC(U , ∂U ) satisfy the following conditions:
(i) deg(Φ, U ) 6= 0;
(ii) dim(F(x)) ≥ n for all x ∈ U where n ≥ 1. Then dim(N (Φ, U )) ≥ n.
P roof. From the assumption (i) it follows that Ø 6= N (Φ, U ) ⊂ U . Suppos-ing the contrary to the conclusion we will have that dim(N (Φ, U )) ≤ n − 1. Then the restrictionΦ = Φ|b N (Φ,U )satisfies the conditions of Lemma 2.2 and hence there exists a continuous selection ϕ : N (Φ, U ) → E of a multifieldb Φb such that 0 6=ϕ(x) for all x ∈ N (Φ, U ). By virtue of the Michael continuousb
The map f : U → E defined by ϕ(x) = x − f (x) is a continuous selection of the multimap F and hence is condensing. It is easy to see that
deg(ϕ, U ) = deg(Φ, U ) 6= 0 and hence N (ϕ, U ) 6= Ø giving the contradiction.
Corollary 2.4. Let B be a closed ball in E; Φ = i − F ∈ DC(B, S) where S = ∂B. If F satisfies the following conditions:
(i) F(x) ∩ B 6= Ø for every x ∈ S;
(ii) dim(F(x)) ≥ n for all x ∈ Int(B) where n ≥ 1. Then dim(N (Φ, B)) ≥ n.
P roof. Condition (i) implies that deg(Φ, B) = 1 ([3], Theorem 1.2.70; [14], Theorem 3.3.2).
3. Topological dimension of the solutions set for some operator inclusions
Let the interval [a, b] be endowed with a Lebesgue measure µ and E be a separable Banach space.
We will need the following property of a measurable multifunction which is the infinite-dimensional version of Lemma 2.6 in [9].
Lemma 3.1. Let Γ : [a, b] → Kv(E) be a measurable multifunction, and suppose that there exists a measurable subset ∆ ⊆ [a, b], µ(∆) > 0 such that dim(Γ(t)) ≥ 1 for every t ∈ ∆. Then for every positive integer m there exists a collection {γi}mi=1 of measurable selections of Γ which are linearly
independent on [a, b].
Now, let F : [0, d] × E → Kv(E) be a multimap satisfying the following properties:
F 1) the multifunction F (·, x) : [0, d] → Kv(E) is measurable for every x ∈ E;
F 3) there exists a function ν(·) ∈ L1
+([0, d]) such that for all x ∈ E:
kF (t, x)k := sup{kyk : y ∈ F (t, x)} ≤ ν(t)(1 + kxk) for a.e. t ∈ [0, d];
F 4) there exists a function k(·) ∈ L1+[0, d] such that for every bounded set D ⊂ E we have that
χ(F (t, D)) ≤ k(t) · χ(D) for a.e. t ∈ [0, d], where χ is the Hausdorff MNC in E.
It is known (see, e.g. [14], Theorem 1.3.4) that for every function x(·) ∈ C([0, d]; E) the multifunction F (t, x(t)) is measurable and hence the super-position multioperator
PF : C([0, d]; E) → P (L1([0, d]; E))
may be defined in the following way:
PF(x) = {f ∈ L1([0, d]; E) : f (t) ∈ F (t, x(t)) a.e. t ∈ [0, d]}.
Let us note that the multioperator PF has the following continuity properties
(see, e.g. [2, 14]).
Lemma 3.2. (i) PF is weakly closed in the following sense: assume the
sequences {xn}∞n=1 ⊂ C([0, d]; E), {fn}n=1∞ ⊂ L1([0, d]; E), fn ∈ PF(xn),
n ≥ 1 are such that xn→ x0, fn→ fw 0. Then f0 ∈ PF(x0); (ii) PF is l.s.c.
Further, consider the operator S : L1([0, d]; E) → C([0, d]; E) having the
form:
S(f ) = z0+ G(f )
where z0 ∈ C([0, d]; E) is a constant function and G : L1([0, d]; E) →
It will be supposed that the operator G satisfies the following conditions. G1) there exists D ≥ 0 such that
kG(f )(t) − G(g)(t)k ≤ D
t Z 0
kf (s) − g(s)k ds
for every f, g ∈ L1([0, d]; E), 0 ≤ t ≤ d;
G2) for any compact K ⊂ E and sequence {fn}∞
n=1 ⊂ L1([0, d]; E) such that
{fn(t)}∞n=1 ⊂ K for a.e. t ∈ [0, d] the weak convergence fn → fw 0 implies
G(fn) → G(f0);
Note that condition (G1) implies that the operator G satisfies the Lipschitz condition
G10) kG(f ) − G(g)k
C ≤ Dkf − gkL1.
We will study the multimap
F = SPF : C([0, d]; E) → P (C([0, d]; E)) and the following operator inclusion:
(1) x ∈ F(x).
Define on bounded subsets of C([0, d]; E) the following MNC ψ with values in the partially ordered set (R2, ≥) where the order is induced by the cone
R2 + of nonnegative pairs: ψ(Ω) = (σ(Ω), modC(Ω)). Here σ(Ω) = sup t∈[0,d] {e−Ltχ(Ω(t))}, L > 0 is large enough, χ is the Hausdorff MNC in E and
Ω(t) = {y(t) : y ∈ Ω}; modC(Ω) = lim
δ→0supx∈Ω|t1max−t2|<δ
is the modulus of equicontinuity of the set Ω. It is easy to see that the MNC ψ is monotone and nonsingular.
We may summarize the facts known from [6] and [14] about the multimap F in the following statement.
Proposition 3.3. Under conditions (F 1) − (F 4) and (G1) − (G2) the mul-timap F has compact convex values, it is continuous and ψ-condensing on bounded subsets.
Let us denote by ΣF the set of all solutions to the operator inclusion (1).
Now we are in position to prove the main result of this section.
Denote a = R0dν(s)ds and r0 = (kz0k + Da) · eDa where ν satisfies
condition (F3) while D satisfies from condition (G1) and z0 = S(0).
Theorem 3.4. Under conditions (F 1) − (F 4) and (G1) − (G2) suppose (F 5) : for some r > r0 the set
∆ = {t ∈ [0, d] : dim(F (t, x)) ≥ 1 f or every x ∈ E, kxk < r} is measurable and µ(∆) > 0. Then the set ΣF is nonempty compact and
dim(ΣF) = ∞.
P roof. Using estimates (G1) and (F3) and applying the standard technique based on the Gronwall type inequality one can see that the set
{x ∈ C([0, d]; E) : x ∈ λ · F(x) for some λ ∈ (0, 1]} is a priori bounded in norm by the constant r0.
Now take a closed ball B = Br(0) ⊂ C([0, d]; E) and consider a family
G : B × [0, 1] → Kv(C([0, d]; E)),
G(x, λ) = λ · F(x).
It is easy to verify that G is ψ-condensing and, as r > r0, it follows that x 6∈ G(x, λ) for all (x, λ) ∈ ∂B × [0, 1]. Now using the property of the homotopy invariance of the topological degree we obtain that
From the assumption (F5) it follows that, taking any x ∈ int(B) and a positive integer n, we have a collection {γi(·)}n+1i=1 of measurable
selec-tions of F (t, x(t)) which are linearly independent on [0, d] (Lemma 3.1). Since the linear operator G is injective, we obtain linearly independent functions {S(γi)}n+1i=1. Now we see that the multimap F satisfies the
con-ditions of Theorem 2.3 and the conclusion dim(ΣF) = ∞ follows from the
arbitrariness of n.
4. Applications: solutions sets of semilinear differential inclusions
(a) Cauchy problem
As an application of the above developed abstract theory we will consider in a separable Banach space E the Cauchy problem for a differential inclusion of the form
(2) x0(t) ∈ Ax(t) + F (t, x(t)), t ∈ [0, d],
(3) x(0) = x0
under the suppositions that the multimap F : [0, d] × E → Kv(E) satisfies conditions (F 1) − (F 4) of the previous section and it is assumed that (A) the linear part A : D(A) ⊂ E → E is the densely defined infinitesimal generator of a C0-semigroup exp{At} .
Recall that the function x(·) ∈ C([0, d]; E), is a mild solution to the problem (2), (3) on the interval [0, d] if it has the following representation
x(t) = exp{At} x0+ Z t
0 exp{A(t − s)} f (s)ds, f ∈ PF(x).
Denote by Σ the set of all mild solutions to the problem (2), (3). It is known (see, e.g. [14]) that under assumptions (A) and (F 1) − (F 4) the set Σ ⊂ C([0, d]; E) is nonempty and compact.
The linear operator G : L1([0, d]; E) → C([0, d]; E) defined as
G(f ) =
Z t
is said to be the Cauchy operator corresponding to the problem (2), (3). It is clear that the set of mild solutions Σ coincides with the fixed points set F ixF where F = S ◦ PF and S(f ) = z0+ G(f ), z0(t) = exp{At} x0.
Lemma 4.1. The operator G satisfies conditions (G1) and (G2). P roof. See [14], Lemma 4.2.1.
Lemma 4.2. The Cauchy operator G is injective.
P roof. Suppose that for certain f ∈ L1([0, d]; E) we have that
(4) v(t) =
Z t
0 exp{A(t − s)} f (s)ds = 0 for all t ∈ [0, d].
Then for any t ∈ [0, d], h > 0 and t + h ∈ [0, d] we have that 0 = v(t + h) − v(t) h = 1 h "Z t+h 0 exp{A(t + h − s)} f (s)ds − Z t 0 exp{A(t − s)} f (s)ds # = 1 h ·Z t 0 exp{Ah} exp{A(t − s)} f (s)ds − Z t 0 exp{A(t − s)} f (s)ds ¸ + 1 h Z t+h t exp{A(t + h − s)} f (s)ds = 1 h(exp{Ah} − I) Z t 0exp{A(t − s)} f (s)ds + 1 h Z t+h t exp{A(t + h − s)} f (s)ds
and therefore, by virtue of condition (4) we have
(5) 1
h
Z t+h
t exp{A(t + h − s)} f (s)ds = 0 for all t ∈ [0, d]
Note now that for every x ∈ E we have that
(6) 1
h
Z t+h
(see, e.g. [16]). Further, we have the following estimation ° ° °1 h Z t+h t exp{A(t + h − s)} (f (s) − x)ds ° ° °≤ M h Z t+h t kf (s) − xkds,
where M = maxt∈[0,d]kexp{At} k.
By virtue of the classical Lebesgue theorem we get
(7) 1
h
Z t+h
t kf (s) − xkds → kf (t) − xk when h → 0
for a.e. t ∈ [0, d]. Note that if x belongs to some countable dense subset ∆ of E we may assume without loss of generality that (7) holds for every x ∈ ∆ and t ∈ m where m ⊆ [0, d] is the set of a full measure. Now take t ∈ m and x ∈ ∆ such that,
kf (t) − xk < ² 4M and choose h1 > 0 such that
¯ ¯ ¯1 h Z t+h t kf (s) − xkds − kf (t) − xk ¯ ¯ ¯< ² 4M for all h ∈ (0, h1). And further choose h2 > 0 such that
° ° °1 h Z t+h t exp{A(t + h − s)} xds − x ° ° °< ² 4 . Now, for h0 = min{h1, h2} and h ∈ (0, h0) we have that
since M ≥ 1. So 1
h
Z t+h
t exp{A(t + h − s)} f (s)ds → f (t) when h → 0 for a.e. t ∈ [0, d].
Taking into account the equality (5) we obtain that f (t) = 0 for a.e. t ∈ [0, d] proving the lemma.
Now we may apply Theorem 3.4 to evaluate the topological dimension of Σ. Theorem 4.3. Under conditions (A) and (F 1)−(F 4) suppose that condition (F 5) holds for r0 = M (kx0k + a)eM a where M = supt∈[0,d]kexp{At}k and
a =R0dν(s) ds. Then dim(Σ) = ∞. (b) Periodic problem
Now we will study the topological dimension of the set of periodic solutions to a semilinear differential inclusion. Let, as before, E be a separable Banach space. For T > 0, let CT(E) denote the space of all continuous T -periodic
functions from R+ into E with the usual sup-norm. Suppose that
A0) A : D(A) ⊂ E → E is a linear operator generating a C
0-semigroup
exp{At} possessing the property that 1 does not belong to the spectrum sp(exp{AT }).
Note that from the above condition it follows that the operator [I − exp{AT }]−1 is well-defined.
It will also be assumed that
FT) the multioperator F : R+ × E → Kv(E) is T -periodic in the first argument:
F (t + T, x) = F (t, x) for all t ∈ R+, x ∈ E.
F : [0, T ] × E → Kv(E), which will be denoted by the same symbol, possesses the properties (F 1) − (F 4) of Section 3 (with the change of d on T ). Therefore, the superposition multioperator PF : C([0, T ]; E) →
P (L1([0, T ]; E) generated by F is well-defined. Note that for x ∈ CT(E), every f ∈ PF(x) will be considered as T -periodically extended on R+.
Let us introduce the multioperator FT : CT(E) → P (C(R+; E)) defined
in the following way:
FTx = {y : y(t) = exp{At} [I − exp{AT }]−1
Z T
0 exp{A(T − s)} f (s)ds
+
Z t
0 exp{A(t − s)} f (s)ds, f ∈ PF(x)}.
From [14], Propositions 6.1.1, 6.1.2 and Theorem 6.1.1 we may easily deduce the following properties of the integral multioperator FT.
Lemma 4.4. (i) The range FT(CT(E)) is contained in CT(E); (ii) ΣT = F ixFT;
(iii) The multioperator FT has compact convex values, it is continuous and can be represented in the form
FT =GPe F
where the linear operator
(8) e G(f )(t) = exp{At} [I − exp{AT }]−1 Z T 0 exp{A(T − s)} f (s)ds + Z t 0 exp{A(t − s)} f (s)ds,
satisfies conditions (G10) and (G2).
To obtain the condensivity of FT we need some additional assumptions. For
a linear operator L : E → E denote by kLk(χ) its χ-norm:
kLk(χ):= χ(LB)
(AF ) the semigroup exp{At} is χ-strongly contractive in the sense that kexp{At} k(χ)≤ e−γt
and the coefficient γ satisfies the estimation γ > 1
T
Z T
0 k(s)ds
where k(·) is the function from the condition (F4). On the space CT(E) consider the MNC
φ(Ω) = (χ(Ω(t)), modC(Ω))
with the values in L∞T × R ordered by the cone K × R+where K is the cone
of a.e. nonnegative functions. It is easy to see that the MNC φ is monotone and nonsingular.
Lemma 4.5. Under conditions (A0), (F
T), (F 1) − (F 4) and (AF ) the
inte-gral multioperator FT is φ-condensing on bounded subsets of CT(E).
P roof. See [14], Theorem 6.1.2.
Lemma 4.6. The linear operatorG is injective.e
P roof. Let G(f )(t) ≡ 0. Taking t = 0 in the formula (8) we obtain thate
[I − exp{AT }]−1 Z T 0 exp{A(T − s)} f (s)ds = 0 and hence e G(f )(t) = Z t 0 exp{A(t − s)} f (s)ds ≡ 0
and Lemma 4.2 yields f (t) = 0 a.e.
Theorem 4.7. Under conditions (A0), (F
T), (F 1) − (F 4) and (AF ) assume
that there exists an a’priori bound q0 for mild T -periodic trajectories of the
family of semilinear inclusions
(9) x0(t) ∈ Ax(t) + λF (t, x(t)), λ ∈ [0, 1]
and suppose that condition (F 5) holds for r0 = q0. Then dim(ΣT) = ∞. Corollary 4.8. Under conditions (A0), (F
T), (F 1)−(F 4) and (AF ) assume
that F satisfies the stronger global estimate:
F 30) there exists a function ν(·) ∈ L1+([0, d]) such that for all x ∈ E : kF (t, x)k ≤ ν(t) f or a.e. t ∈ [0, T ]
(instead of condition (F 3)) and suppose that condition (F 5) holds for r0 = M aT (M k[I − exp{At}]−1k + 1) where M = maxt∈[0,T ]kexp{At}k and
a =R0T ν(s)ds. Then dim(ΣT) = ∞.
P roof. It is sufficient to verify that the number r0 gives the a priori bound for T-periodic solutions of the family (9).
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